Subgroups — #26242806

Order 12032.

764 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244329	C₂	1	2	yes	maximal, complemented
2	#20244003	C₂	1	4	yes	maximal, complemented
3	#20243681	C₂	1	2	yes	maximal
4	#20243568	C₂	1	4	yes	maximal, complemented
5	#20243505	C₂	1	2	yes	maximal, complemented
6	C₄×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	#15688878	C₂²	1	4	yes	—
15	#15688878	C₂²	1	4	yes	—
16	#15688878	C₂²	1	4	yes	—
17	C₂×#124×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#391	C₄	1	1	yes	characteristic
21	C₄₇×#390	C₄	1	1	yes	characteristic
22	C₄₇×#388	C₄	1	4	yes	—
23	C₄₇×#382	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	—	4	4	no	—
27	C₂²×C₄₇⋊C₈	—	4	2	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
35	#13093935	—	4	2	no	—
36	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
39	#124×C₄₇	—	2	4	no	—
40	C₄×Q₈×C₄₇	—	4	1	no	—
41	C₄×Q₈×C₄₇	—	4	1	no	—
42	C₄×D₄×C₄₇	—	4	1	no	—
43	C₄×D₄×C₄₇	—	4	1	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	#95×C₄₇	C₂×C₄	1	4	yes	—
57	#95×C₄₇	C₂×C₄	1	4	yes	—
58	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	—	4	1	no	—
61	C₂×Q₈×C₄₇	—	4	1	no	—
62	C₂×D₄×C₄₇	—	4	1	no	—
63	C₂×D₄×C₄₇	—	4	1	no	—
64	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₄₇	—	4	1	no	—
78	C₄⋊C₄×C₄₇	—	4	1	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₂²⋊C₄×C₄₇	—	2	8	no	—
81	C₂²⋊C₄×C₄₇	—	2	8	no	—
82	C₂²⋊C₄×C₄₇	—	4	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	2	1	no	—
86	C₄²×C₄₇	—	2	1	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	4	1	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	4	8	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
104	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	Q₈×C₄₇	—	4	4	no	—
106	D₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₄₇	#131	1	4	yes	—
128	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
132	C₄₇⋊C₈	—	4	4	no	—
133	C₄₇⋊C₈	—	4	4	no	—
134	#23352	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	4	no	—
136	C₂²×C₄₇	—	2	8	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	—	4	2	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	#518	1	1	yes	characteristic
141	C₂²×C₄₇	#520	1	2	yes	—
142	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#131	1	2	yes	—
144	C₂²×C₄₇	#568	1	1	yes	characteristic
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	4	2	no	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	—	4	1	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	—	4	2	no	—
152	C₄×C₄₇	H(3,𝔽₄)	1	1	yes	characteristic
153	C₄×C₄₇	#562	1	2	yes	—
154	C₄×C₄₇	#563	1	1	yes	characteristic
155	C₄×#124	D₄₇	1	1	yes	characteristic
156	#1985	—	47	2	no	—
157	#1924	—	47	4	no	—
158	#1869	—	47	2	no	—
159	#1850	—	47	4	no	—
160	#1831	—	47	2	no	—
161	C₂×C₄₇	—	4	4	no	—
162	C₂×C₄₇	#2379	1	1	yes	characteristic
163	C₂×C₄₇	#2973	1	1	yes	characteristic
164	C₂×C₄₇	#2979	1	1	yes	characteristic
165	C₂×C₄₇	#2990	1	2	yes	—
166	C₂×C₄₇	#2992	1	2	yes	—
167	C₂³×C₈	—	47	2	no	—
168	C₂×#124	Dic₄₇	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	47	2	no	—
172	C₂×C₄⋊C₈	—	47	2	no	—
173	C₂×C₄⋊C₈	—	47	2	no	—
174	C₂×C₂²⋊C₈	—	47	4	no	—
175	C₂×C₂²⋊C₈	—	47	2	no	—
176	C₂×#97	—	47	2	no	—
177	#391	Dic₄₇	1	1	yes	characteristic
178	#390	Dic₄₇	1	1	yes	characteristic
179	#388	Dic₄₇	1	4	yes	—
180	#382	Dic₄₇	1	1	yes	characteristic
181	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
182	C₄³	C₂×D₄₇	1	1	yes	characteristic
183	#336	—	47	4	no	—
184	#336	—	47	4	no	—
185	#336	—	47	4	no	—
186	C₄₇	#23352	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
188	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	47	4	no	—
192	C₂²×C₈	—	188	2	no	—
193	C₂²×C₈	—	188	4	no	—
194	#124	—	2	4	no	—
195	C₄×Q₈	—	4	1	no	—
196	C₄×Q₈	—	4	1	no	—
197	C₄×D₄	—	4	1	no	—
198	C₄×D₄	—	4	1	no	—
199	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
202	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
203	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
205	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
210	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
211	C₄⋊C₈	—	188	2	no	—
212	C₄⋊C₈	—	188	2	no	—
213	C₄⋊C₈	—	188	2	no	—
214	C₂²⋊C₈	—	188	2	no	—
215	C₂²⋊C₈	—	188	4	no	—
216	#97	—	188	2	no	—
217	#95	C₂×Dic₄₇	1	4	yes	—
218	#95	C₂×Dic₄₇	1	4	yes	—
219	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	#10625329	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
234	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	94	4	no	—
240	C₂×C₈	—	94	4	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	188	8	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×#10625329	1	4	yes	—
265	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	D₄	—	4	4	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂×#10625329	1	2	yes	—
288	C₂×C₄	#13094031	1	4	yes	—
289	C₂×C₄	#13094069	1	1	yes	characteristic
290	C₂×C₄	#13094069	1	1	yes	characteristic
291	C₂×C₄	#13094070	1	2	yes	—
292	C₂×C₄	#13094071	1	2	yes	—
293	C₈	—	188	4	no	—
294	C₈	—	188	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	8	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	4	no	—
300	C₂²	#15689598	1	1	yes	characteristic
301	C₂²	#15689606	1	2	yes	—
302	C₂²	C₂²×#10625329	1	1	yes	characteristic
303	C₂²	C₂×#13094031	1	2	yes	—
304	C₂²	#15689890	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	#15689865	1	1	yes	characteristic
313	C₄	#15689867	1	2	yes	—
314	C₄	#15689873	1	1	yes	characteristic
315	C₂	—	4	4	no	—
316	C₂	#20246562	1	1	yes	characteristic
317	C₂	#20252417	1	1	yes	characteristic
318	C₂	#20252455	1	1	yes	characteristic
319	C₂	#20252513	1	2	yes	—
320	C₂	#20252527	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#124×C₄₇	C₂
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#124	D₄₇
hypocenter	C₄₇	#23352
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20244329	C₂	semidirect
2. #20244003	C₂	semidirect
4. #20243568	C₂	semidirect
5. #20243505	C₂	semidirect
186. C₄₇	#23352	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20244329	2
#20244003	2
#20243681	2
#20243568	2
#20243505	2
C₄×#124×C₄₇	2
#23352	47